Abstract
Let M be a module over a commutative ring R and U a nonempty proper subset of M.
In this paper, the extended total graph, denoted by ET_{U}(M), is presented, where U is a
multiplicative-prime subset of M. It is the graph with all elements of M as vertices, and for distinct m,n\in M, the vertices
m and n are adjacent if and only if rm+sn\in U for some r,s\in R\setminus (U:M). We also study the two (induced) subgraphs ET_{U}(U) and ET_{U}(M\setminus U), with vertices U and M\setminus U, respectively. Among other things, the diameter and the girth of ET_{U}(M) are also studied.